Appendix The Project Proposal for Funding Department · PDF fileThe Project Proposal for Funding Department of Mathematics Addis Ababa University, ETHIOPIA August 31, 2010 ... { Compile

Appendix

to

The Project Proposal for Funding

Department of Mathematics

Addis Ababa University, ETHIOPIA

August 31, 2010

ApplicantSemu Mitiku, Dr.

Department of Mathematics,

Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

Tel. 00251 11 123 9461(Office); 00251 911 744898 (cell)

E-mail: [email protected] OR [email protected]

Project Title: Capacity Building in Research and Grad-uate Education in Mathematics in Ethiopia

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Component I: Analysis

Proposed Area of Research for PhD Dissertation in Analysis

Seid MohammedDepartment of Mathematics, Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

E-Mail: [email protected]

1.1 Background

In function theory there are a number of inequalities (sharp) involving certain quan-

tities which describe the growth of meromorphic functions (or � -subharmonic func-

tions) in the plane .In this direction one can investigate the behavior of those functions

for which equality holds in the inequality under consideration .Problems of this nature

date back to the 19th century. For example Valiron(1914) and Wiman (1915)proved

that

limr→∞

supA(r, u)

B(r, u)≥ cos ��

where

A(r, u) = inf u(z) B(r, u) = supu(z) , on ∣z∣ = r , u(z) = log ∣f ∣

and f is an entire function of order � for 0 < � < 1. The inequality is sharp.

Drasin and Shea (1969) proved certain regularity theorem of those functions for which

equality holds in the above inequality.

In this project we consider the following three problems: two extremal problems the

specific objectives of which are to find a global behavior (regularity) of T (u, r) , N(u, r)

2

and u(rei�) and other related quantities as r → ∞; and one problem on differential

equations.

1.2 Problem Statement

Let u = u1 − u2 be a� -subharmonic function in the complex plane, where u1 and u2are subharmonic functions. We write

N(r, u) =1

2�

2�∫0

u(rei�)d�.

The Nevanlinna characteristic T (r, u) of u is defined by

T (r, u) = N(r, u+) +N(r, u2)

and the order � of u by

� = limlog T (r, u)

T (r, u)

If � is finite, it is well known that T (r, u) has sequence of Polya Peaks {rn} of order�,

i.e, a sequence {rn} and {�n} satisfying �n → 0 and �nrn →∞ as n→∞ and

T (r, u) ≤ (1 + �n)

(r

rn

)T (r, u) , (�nrn ≤ r ≤ rn

�n)

Let {rn}∞n=1 be a sequence of Polya peaks of order of T (r, u). Set

�n =1

2m{� ∈ (�, �] : u(rne

i�) > 0}

(n = 1, 2, ...)

where m is Lebesgue measure. Let �0 = inf �n and � be the smallest nonnegative

real number such that

cos�� = 1− � (0.0.1)

where

� = 1− lim supr→∞

N(r, u2)

T (r, u)(0.0.2)

called the Nevalinna deficiency of u.

3

A. Baernstein (1973) proved Edrei’s spread conjecture:

�0 ≥ min{�, 1

2cos−1(1− �)} (0.0.3)

We say the spread is extremal if

�0 = �. (0.0.4)

If u is subharmonic, � = �2�

for � > 12.

1. Problem I:

If 0 < � < � and if (0.0.4) holds Edrei proved certain regularity theorems

in case u = log, where f(z) is a meromorphic function. In this direction one

can investigate specifically the asymptotic behavior of the class of subharmonic

functions of order greater than 1/2 which satisfies (0.0.4). A typical example

of subharmonic function which satisfies (0.0.4) is given by

u(rei�) =

{��r� cos�� if ∣�∣ < �

2�,

0 if ∣�∣ < �2�

.

2. Problem II: Let u = u1 − u2 be a �-subharmonic function and put A(r) =

inf∣z∣=r

u(z). Essen, Rossi and Shea have proved that under suitable conditions

lim supr→∞

A(r, u)

T (r, u)≥ −�� sin�(� − �) (0.0.5)

where T (r, u) is the Nevanlinna characteristic of u; � is the order of u and 0 <

� < 1, and as defined in (0.0.1). The inequality is sharp. One can investigate

the asymptotic behavior of the class of functions u for which equality holds in

(0.0.5), i.e, the behavior of extremal functions .

3. Problem III. Another possible research area would be to consider problems in

differential equations where substantial knowledge of real analysis is required.

One can investigate:

(a) Dirichlet problems of elliptic PDEs in bounded or un-bounded domains.

More specifically this involves issues related to

∙ Existence of solutions

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∙ Uniqueness of solutions

∙ Asymptotic boundary behavior of solutions

∙ Regularity of solutions

(b) Qualitative properties of solutions of Elliptic PDEs. Including, but not

limited to,

∙ Maximum-Minimum principles and comparison principles

∙ Harnack Inequalities

∙ Cauchy-Liiouville’s properties

∙ Symmetry properties

1.3 Enrollment of PhD Students

There will be three junior staff members joining the PhD program of under

this project: one will be enrolled under a sandwich PhD program under the

supervision of Prof.Richard Bogvard and the remaining two will be enrolled

under in-house PhD program.

1.4 Activity plan for the years 2010-2013

1.4.1 For student under Sandwich PhD program

∙ Year I (2010)

– Enrollment of the junior staff members interested in the field of anal-

ysis and differential equations (begins September 2010)

∙ Year II (2011)

– Students who have completed successfully all the requirements as per

stated in the curriculum will undertake formulation of problem in

collaboration with their advisor (s).

– Student will visit supervisor in Sweden from Jan. 1, 2011 to Feb. 1,

2011.

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– Student will visit supervisor in Sweden from Sept. 1, 2011 to Dec. 31,

2011.

– Literature review

∙ Year III (2012)

– Student will continue research work in Sweden for six months from

January 1, 2012 to June 30, 2012.

– participate in conference

∙ Year IV (2013)

– Supervisors abroad visit PhD student in Ethiopia in Oct 2012 where

the students presents a progress report and conduct a seminar at de-

partmental level on his/her findings.

– Prepare the results to be submitted to a journal for publication.

– Compile the PhD thesis or dissertation.

– Supervisor visit Ethiopia for defense of the students on December

2013.

1.4.2 For students In-house PhD Program

∙ Year I (2010)

– Enrollment of the junior staff members interested in the field of analysis

and differential equations (begins September 2010).

∙ Year II (2011)

– Students who have completed successfully all the requirements as per

stated in the curriculum will undertake formulation of problem in col-

laboration with their advisor (s).

– Literature review

∙ Year III (2012)

– Will continue their research work with their local advisor.

– Present progress report and conduct seminars at departmental level.

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– Participate in conference

∙ Year IV (2013)

– Present progress report and conduct seminars at departmental level.

– Prepare the results to be submitted to a journal for publication.

– Compile the PhD thesis or dissertation.

– Defend his/her thesis on December 2013.

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Component II: Differential

Equations

Analysis of Boundary-Domain Integral (Integro-Differential) Equation

systems and Qualitative Theory of Differential Equations

Involving Researchers:

Tsegaye Gedif and Taddese AbdiDepartment of Mathematics, Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

E-Mail: [email protected]

2.1 Purpose and significance of the activity

Many universities have been established in recent years in several regions of Ethiopia

and a few have been started Masters degree programs. However, with possible excep-

tion of the Addis Ababa University, these universities lack qualified mathematicians

trained at the PhD level. To address this critical problem, the Mathematics Depart-

ment of Addis ababa University has launched a full-fledged in-house PhD program.

Therefore, this project

∙ will help to enhance the quality and the academic level of student advising

and pedagogy in differential equations; and strengthen the academic standard

of the program by expanding knowledge in this field of specialization through

collaboration with scientists in United States and Europe.

8

∙ intend to contribute to the development of relevant methods and choice of real-

istic models in the analysis of biological, ecological and environmental problems

in the nation.

2.2 Background

2.2.1 Analysis of Boundary-Domain Integral (Integro-Differential)

Equations:

Partial Differential Equations (PDEs) with variable coefficients often arise in math-

ematical modelling of inhomogeneous media (e.g. functionally graded materials or

materials with damage induced inhomogeneity) in solid mechanics, electromagnetics,

thermo-conductivity, fluid flows through porous media, and other areas of physics

and engineering.

Generally, explicit fundamental solutions are not available if the PDE coefficients are

not constant, preventing formulation of explicit boundary integral equations for them,

which can then be effectively solved numerically. Nevertheless, for a rather wide class

of variable-coefficient PDEs it is possible to use instead an explicit parametrix (Levi

function) taken as a fundamental solution of corresponding frozen-coefficient PDEs,

and reduce Boundary Value Problems (BVPs) for such PDEs to explicit systems of

Boundary-Domain Integral Equations (BDIEs), see e.g. [Mi02, CMN09, Mi06] and

references therein. However this (one-operator) approach does not work when the

fundamental solution of the frozen-coefficient PDE is not available explicitly (as e.g.

in the Lame system of anisotropic elasticity).

To overcome this difficulty, one can apply the so-called two-operator approach, formu-

lated in [Mi05] for some non-linear problems, that employs a parametrix of another

(second) PDE, not related with the PDE in question, for reducing the BVP to a

BDIE system. Since the second PDE is rather arbitrary, one can always chose it

by such a way, that its parametrix is available explicitly. A simplest choice for the

second PDE is the one with a fundamental solution explicitly available.

To analyse the two-operator approach we apply in [AM10] one of its linear versions to

the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-

coefficient PDE reducing it to four different BDIE systems. Although the considered

9

BVP can be reduced to (other) BDIE systems also by the one-operator approach,

it can be considered as a simple ”toy” model showing the main features of the two-

operator approach to appear also in reducing more general BVPs to BDIEs. The

two-operator BDIE systems are nonstandard systems of equations containing integral

operators defined on the domain under consideration and potential type and pseudo-

differential operators defined on open sub-manifolds of the boundary.

In this research, developing further the results of [Mi05, CMN09, AM10], we plan to

give a rigorous analysis of the one-operator and two-operator BDIEs and show that

the BDIE systems are equivalent to the mixed BVP and thus are uniquely solvable,

while the corresponding boundary domain integral operators are invertible for more

general PDEs in appropriate Sobolev-Slobodetski (Bessel-potential) spaces.

2.2.2 Qualitative Theory of Differential Equations:

Delay Differential Equations (DDEs) which are a generalization of ordinary differen-

tial equations (DDE with all delays zero) have long played a significant role in the

study of ecological models (population dynamics), in the development of models of

control systems and models of chemical processes etc. Most of the systems (Biolog-

ical, Chemical, Engineering etc) have inherent time lag. A better way of describing

the property of such systems is by use of differential equations that give significant

account of initial history of the system and hence the need for differential equations

with time lag, (DDE). The logistic model (P.F. Verhulst) presumes that the repro-

ductive rate of a population is instantaneous. However, in reality there is a time

lag on the order of a generation of organisms as described by delay logistic equation

(G.E. Hutchinson).

Delays persist in vehicular traffic flow where the follower takes some time (reaction

time) to respond to a stimulus from the lead car. Thus DDE is nowadays becom-

ing a centerpiece of realistic models in almost all areas of research in science and

engineering.

DDEs problems are oftentimes nonlinear and the solutions exhibit oscillatory behav-

ior (delay-induced oscillation). Mathematical tools for solving DDE problems involve

approximation techniques. In this regard one can mention asymptotic methods (an-

alytical) and numerical methods for DDEs, i.e. DDE solvers (dde23, xppout etc).

10

Recent studies of qualitative property of differential equations with delay which in-

cludes stability (S.N. Chow) and asymptotic behavior (Burton-Furumochi) of equi-

librium solutions rests on fixed point theorems. The relevant numerical methods are

designed to deal with discontinuities especially in the lower order derivatives, unlike

their counter parts for ODE that generally work with continuous derivatives. Hence,

the propagation of discontinuity in the solution which is a typical characteristic of

DDEs needs to get special attention.

2.3 Objectives

2.3.1 Boundary-Domain Integral (Integro-Differential) Equa-

tions:

2.3.1.1 General Objectives:

The general objectives of this research are aimed at localized boundary-domain inte-

gral and integro-differential for linear and nonlinear BVPs with variable coefficients.

It includes

1. reduction of Boundary Value Problems (BVPs) for linear and nonlinear PDEs

with variable coefficients to Boundary-Domain Integral Equations (BDIEs) or

Boundary Domain Integro-Differential Equations (BDIDEs) with localized sup-

port; and

2. analysis of BDIEs (BDIDEs), that is, to show that the newly obtained BDIE

(BDIDE) systems are equivalent to the original BVPs and thus are uniquely

solvable, while the corresponding boundary domain integral operators are in-

vertible.

2.3.1.2 Specific Objectives:

Specific objectives that should be reached within the period of support applied for

include: reduction of BVPs on the domain to BDIEs on the boundary of the do-

main using one-operator and two-operator approaches and their analysis for scalar

11

elliptic PDEs, elliptic PDE systems of 2nd order and linear scalar PDEs with matrix

coefficients. That is, to show that the Boundary-Domain Integral Equation sys-

tems are equivalent to the original mixed BVP and thus are uniquely solvable, while

the corresponding boundary domain integral operators are invertible in appropriate

Sobolev-Slobodetskii (Bessel-potential) spaces. Namely,

1. Analysis of one-operator direct Boundary Domain Integral Equations, re-doing

the paper [CMN09] for 2-dimentional domains.

2. Analysis of the two-operator Boundary Domain Integral Equations, generalizing

results in [AM10] for the cases b = b(y) in the auxiliary operator (0.0.11), and

show that the case b = a(y) will follow from this generalization.

3. Analysis of two-operator BDIEs for elliptic PDE systems of 2nd order.

4. Analysis of two-operator BDIEs for linear scalar PDEs with matrix coefficients.

2.3.2 Qualitative Theory of Differential Equations:

2.3.2.1 General Objectives:

Our aim is to compare and use new analytical techniques (many), numerical schemes

in the Department of Mathematics and advancing across the College of Natural Sci-

ences, Addis Ababa University that hosts several interdisciplinary graduate programs

and beyond. In particular, the focus is on graduate training at masters’ level with

research based M.Sc.

2.3.2.2 Specific Objectives:

We want to concentrate on the investigation of eventually positive solutions. Finding

extensions of existence theorems for non-oscillatory solutions of DDEs and criteria

for boundedness of positive solutions of such problems is the major concern.

12

2.4 Strategy and Plan

2.4.1 Boundary-Domain Integral (Integro-Differential) Equa-

tions:

By strengthening the existing collaborations and initiating new ones with scientists

in Europe and United States, a joint supervision of two PhD students is planned

under this project. Results in [Mi05, CMN09, AM10] are to be developed further.

The planned project can be completed with in 3-4 years as follows:

2.4.1.1 Prior Research:

In [AW09], a nonzero linear differential operator L(D) with constant coefficients

L(D) =∑

0≤∣�∣≤2

a�D�, (0.0.6)

where K is a finite set in the space of multi-indices ℕn0 = ℕ0×ℕ0×⋅ ⋅ ⋅×ℕ0(n-copies)

with ℕ0 = ℕ∪{0}. For � ∈ ℕn0 , ∣�∣ = �1 +�2 + ⋅ ⋅ ⋅+�n, max ∣�∣ = m, a� ∈ K are

constants; and D� = D�11 ×D�2

2 ×⋅ ⋅ ⋅×D�nn , where D

�jj = 1

i∂�j

∂x�jj

is considered. And a

criterium for the condition of partial hypoellipticity of (0.0.6) in terms of fundamental

solutions is given.

The notion of fundamental solutions studied in [AW09] served as a tool to understand:

1. the reduction of Boundary Value Problems (BVPs) with constant coefficients

in a domain to Boundary Integral Equations (BIEs) on the boundary of the

domain using fundamental solution,

2. the reduction of Boundary Value Problems with variable coefficients in a do-

main to Boundary Domain Integral Equations (BDIEs) on the boundary of the

domain using parametrix instead of fundamental solution in [CMN09] and

3. the analysis of Boundary Domain Integral Equations in [CMN09].

13

And in [AM10], the mixed (Dirichlet-Neumann) type Boundary Value Problem (BVP)

Lau = f in Ω (0.0.7)

u+ = '0 on ∂DΩ (0.0.8)

T+a u = 0 on ∂NΩ (0.0.9)

where '0 ∈ H12 (∂DΩ), 0 ∈ H−

12 (∂NΩ) and f ∈ L2(Ω), for a second-order scalar

elliptic differential equation

Lau(x) := La(x, ∂x)u(x) :=3∑i=1

∂

∂xi

[a(x)

∂u(x)

∂xi

]= f(x), x ∈ Ω, (0.0.10)

with variable coefficients; where u is unknown function and f is a given function in

an open three-dimensional region Ω of ℝ3, is considered.

Taking another auxiliary linear elliptic partial differential operator Lb such that

Lbu(x) := Lb(x, ∂x)u(x) :=3∑i=1

∂

∂xi

[b(x)

∂u(x)

∂xi

], (0.0.11)

where b ∈ C∞(ℝ3), b(x) > 0; and applying the two-operator approach formulated

in [Mi05], the problem is reduced to Boundary Domain Integral Equations (BDIEs).

Using the results of [CMN09], equivalence of the boundary domain integral equa-

tions to the mixed BVP, their solvability and invertibility of associated operators are

analyzed in appropriate Sobolev spaces.

2.4.1.2 Research Plan:

In this research, we plan to develop further the results in [Mi05, CMN09, AM10], give

a rigorous analysis of the one-operator and two-operator BDIEs and show that the

BDIE systems are equivalent to the mixed BVP and thus are uniquely solvable, while

the corresponding boundary domain integral operators are invertible in appropriate

Sobolev-Slobodetski (Bessel-potential) spaces.

∙ Year 2011:

1. Integral Representation for solutions of elliptic operators using Green’s

function will be revised.

14

2. Reduction of Boundary Value Problems (BVPs) with constant coefficients

in a domain to Boundary Integral Equations(BIEs) on the boundary of

the domain using fundamental solutions, and basic solution methods be

studied.

3. One-operator approach formulated in [CMN09] will be studied. That is,

(a) Boundary Value Problems (BVPs) with variable coefficients in a do-

main will be reduced to Boundary-Domain Integral Equations (BDIEs)

on the boundary of the domain using parametrices.

(b) Equivalence of BDIE systems to the original mixed BVP will be shown.

(c) Unique solvability of BDIE systems will be proved.

(d) Invertibility of the corresponding boundary domain integral operators

will be shown.

4. Results in [AM10] will be generalized for the cases b = b(y) in the auxiliary

operator (0.0.11), and the case b = a(y) will follow from this generalization.

(a) The new result will be presented in international conferences, feed-

backs (if any) will be accommodated.

(b) Results will be formulated and prepared for publication.

∙ Year 2012:

1. Formulation of one-operator BDIEs, for 2-dimensional domains, that is,

investigating the results in [CMN09] for n = 2.

2. Formulation and analysis of two-operator approach for elliptic PDE sys-

tems of 2nd order.

(a) Results will be presented in seminars, national and/ or international

conferences.

(b) Feedbacks (if any) will be accommodated, and results will be formu-

lated and prepared for publication.

∙ Year 2013:

1. Analysis of one-operator BDIEs for 2-dimensional domains, that is, re-

doing the paper [CMN09] for n = 2.

15

(a) Boundary Value Problems (BVPs) with variable coefficients in a do-

main will be reduced to Boundary-Domain Integral Equations (BDIEs)

on the boundary of the domain using parametrices.

(b) Equivalence of BDIE systems to the original mixed BVP will be shown.

(c) Unique solvability of BDIE systems will be proved.

(d) Invertibility of the corresponding boundary domain integral operators

will be shown.

(e) Results will be presented in seminars, national and/ or international

conferences.

(f) Feedbacks (if any) will be accommodated, and results will be formu-

lated and prepared for publication.

2. Formulation and analysis of two-operator approach for linear scalar PDEs

with matrix coefficients

(a) Results will be presented in seminars, national and/ or international

conferences.

(b) Feedbacks (if any) will be accommodated, and results will be formu-

lated and prepared for publication.

3. PhD candidates will be recommended for defense.

4. New research directions for more general PDEs will be identified.

2.4.2 Qualitative Theory of Differential Equations:

A student taking part in this training will write a thesis on his findings in the end.

∙ Year 2011

The following are activities planned for year I.

1. Asses the required facility for availability and adequacy

2. Collect and organize current and relevant materials

3. Identify a would be graduate student

4. Formulation of the problem

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∙ Year 2012

Activities planned for year II are as follows

1. Developing a criteria (sufficient and necessary condition) for boundedness

of positive solutions

2. Obtain an extension of existence theorems for nonoscillatory solutions.

3. Conduct numerical experiment

4. Write and defend thesis, M.Sc.

∙ Year 2013

This is the final year of the plan and the activities for year III are

1. Identify further directions of numerical treatment of prototype model prob-

lem and suggest possible improvements.

2. Present the findings in a conference, national or international.

3. Write a proposal for PhD work.

2.5 Expected outcomes, impact and dissemination

Knowledge gained within this research activity will be evaluated and disseminated

through publication in journals and reports at conferences; through postgraduate

programs in general and through training and advising students of newly launched in-

house PhD program in mathematics in particular; by arranging courses or workshops

on national or regional level or by receiving students for training. The new findings

and improvements made on existing methods and techniques will be considered for

publication in a journal. This result will be used as references, resource material by

graduate students in relevant discipline in the college.

2.6 Collaboration with other Scientists

To keep the competence and commitment for success of the in-house PhD program

in mathematics, it is crucial that the faculty engaged in the program keep themselves

abreast of the latest advances in their fields. Therefore, strengthening the existing

17

collaborations and initiating new ones with scientists in Europe and USA will lessen

the adverse effect of having to work in an environment that lacks research facilities,

resource and a core group of active researchers in their fields.

Thanks to the MARM (Mentoring African Research in Mathematics) project funded

by LMS/Nuffield Foundation, the collaboration between Prof. Sergey E. Mikhailov

of Brunel University, UK and the research group led by Dr. Tsegaye G. Ayele of

Addis Ababa University established in 2006, succeeded to achieve the project goals

including publication of two papers [AW09, AM10] and one on preparation.

Regardless of the termination of the pilot project in 2008, Prof. Sergey E. Mikhailov

is still collaborating actively with the research group, supporting the in-house PhD

program launched by the Department of Mathematics, Addis Ababa University and

willing to co-advise 2 (two) PhD students. The research group, therefore strengthens

this collaboration and initiates new others so as to enhance the quality and the

academic level of student advising and pedagogy in differential equations.

2.7 Postgraduate Students

Graduates with M.Sc degree from different universities of the country who are joining

the in-house PhD program in mathematics at Addis Ababa University, are required

to take some course before the work on their dissertation stars. Even though most

of the courses are planned to be offered at home university, it may require to invite

collaborating professors from Europe, USA and Africa to give compact courses re-

lated to specific research direction. And the PhD students could be recommended to

take courses offered at collaborating universities, participate in schools, workshops

and conferences at national, regional or international level. Therefore postgraduate

students and their supervisors are dependent on ISP funding for the normal running

of the PhD program.

For the two accepted PhD students, projects consisting of the following problems are

suggested and the obtained results would be publishable.

18

1. Analysis of one-operator BDIEs for 2-dimensional domains, that is, re-doing

the paper [CMN09] for 2-dimensional domains. It is not always trivial since

the parametrix is different and the single layer potential can be non-invertible

and one should work around this as people do in Boundary Integral Equations

(BIEs) which seems to be feasible for PhD students.

2. Formulation and analysis of two-operator approach for elliptic PDE systems of

2nd order.

3. Formulation and analysis of two-operator approach for linear scalar PDEs with

matrix coefficients.

19

Component III: “Multilevel

Programming”

Involving Researchers:

Semu Mitiku and Berhanu GutaDepartment of Mathematics, Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

E-Mail: [email protected] OR [email protected]

Keywords: Multilevel programming, hierarchical planning, optimization.

3.1 Introduction

Decision making is an integral part of management, planning, controlling and moti-

vation processes. The decision maker selects one strategy over the others depending

on some criteria. During any process in economy, planning and management the

problem of taking decisions and to find the best decision in a certain sense is one

of the most important tasks in human activities. A good decision is the process of

optimally achieving a given objective. It is obvious, that each decision can be done in

every case subject to certain restrictions (states) and each decision has consequences

for the further development of the concerning process. Also obviously the number of

influences to the process at the beginning (initial position) and during the process can

be very large (also infinity). Therefore, it is very important to have a scientific deci-

sion making process which keep the risks of decisions small and makes the successful

run of the process possible. Such scientific decision making process is characterized

by an adoption of systematic, logical and thorough-going reasoning to understand

20

the problem situation and to arrive at the best possible decisions through a process

of abstraction, observation, experimentation, measurement and analysis of data in

the relevant real world decision environment.

The design of large and complex systems involve decomposition of the system into

a number of smaller subsystems, each with its own goals and constraints. One of

the most common interconnections between such systems is the hierarchical form. In

hierarchical systems a decision maker at one level controls or coordinates the decision

makers on the levels below it and in turn it is controlled by the decision maker on the

level above it. Moreover, it is assumed that a decision maker at each level has a certain

degree of autonomy, i.e., he/she has the freedom to choose the best options among

the alternatives in his/her domain (or control area) and has a different objective

(possibly not in harmony with objectives in the other levels). A system decomposed

in this manner is referred to as a decentralized multi-level system, (Mesarovic 1970

[8]).

In decentralized decision systems, one decision maker can influence the outcome and

decisions of others, and thereby improve his/her own objective. Planning in such

environment has long been recognized as an important decision making problem as

described in Bialas and Karwan (1979) [1].

The allocation of resources by a national government to lower level budget units,

which have a certain degree of their own autonomy, has a multi-level and hierarchical

structure. One level of government unit distributes resources to lower levels, and this

similar process goes further down to some number of levels. For instance, the federal

government of Ethiopia allocates a certain amount of budget to Universities in the

country, and decides on the policy to be materialized by them. Each University after

receiving the budget in turn allocates (or decides on the amount of) budget to its

faculties (of course part of the budget could be used by the central administration of

the University itself) and pass decisions on certain matters that the faculties should

consider. The third level of budget and decision units, the faculties decide on what

to do and how to effectively use the budget. It is obvious that every reaction of the

lower level unit affects the value of the upper level and that of the middle level also

affects both upper and lower levels. We take into account the fact that decisions

about resource usage at the sub-levels cannot be controlled (although they may be

predictable) once resources have been allocated.

Mathematical programming models to solve problems of the above type have been

21

studied since 1960s, (Dantzig and Wolfe 1960 [6]).

In decentralized decision systems, one decision-maker can influence the decisions of

others, and thereby improve his own objective. The primary objective of each level

is to maximize its benefit from the system. The elements of the general model are

decision makers responding to external information. Using this information as an

input the decision maker formulates a response. This produces an environment in

which each decision maker is interacting with, and influenced by the outputs of, other

decision makers.

Many resource allocation or planning problems require compromises among the ob-

jectives of several interacting individuals or agencies, most of the time, arranged in

hierarchical administrative structure and can have independent even sometimes con-

flicting objectives. A planner at one level of the hierarchy may have its objective

function determined partly by variables controlled at other levels. Assuming that the

decision process has a preemptive nature and having r levels of hierarchy, we consider

the decision maker at level r to be the leader and those at lower levels to be followers.

Ethiopia have a federal state political structure and resource allocation to each budget

unit and measuring the efficiency of each unit will take a hierarchical form. Allocating

its meager resource efficiently and measuring the efficiency of its outcome is one of

the major problem at the Federal government level.

In this study we consider a decentralized multilevel hierarchical system, where there

is one higher level decision maker, called the leader, and lower level decision makers

(called followers) where there are more than one decision makers at each level. The

leader will take a preemptive decision to control the decision makers at lower levels.

Mathematically, modeling and studying the behavior of reactions at each level and

the optimal allocation of resources (and setting policies) will be studied.

3.2 Objectives

∙ To study the problems that arise in allocating resources and measuring efficiency

of each unit in multi-level decision making organizations.

∙ To propose a better solution method for the problem

22

∙ To summarize existing solution methods for multi-level programming problems

and propose working algorithms for general tri-level programming problems.

∙ To develop a software which supports the decision making strategy in optimally

allocating resources and measuring the efficiency of unit.

∙ To train Masters level as well as Ph.D. level students specializing in the area.

3.3 Proposed Way of Conducting the Research

The first phase of the study is more of developing theoretical background and re-

viewing literature. In this regard, the plan is to first, recruit bright students for

the Masters and PhD programs to work as a team and institute a weekly research

seminar serious dedicated to the research topic. Then, the participating students can

have a good background in the area and will review existing literature relevant to the

topic. We also need to visit some institutions which have hierarchical decision mak-

ing structures to understand their practical difficulties in their day-to-day activities,

thereby collecting necessary data for the theoretical studies.

Bi-level programming problem is the most researched topic in the multilevel opti-

mization field. However, most of the algorithms developed to solve bi-level problems

assume convexity of the lower level problems. We will start by formulating a general

framework that help to solve bi-level problems with non-convex lower level problems.

This will give us a way of reformulating and solving tri-level problems using the

bi-level algorithm approach.

The final phase of the research will be implementation of the algorithms as a tool for

decision support system. Since size of the problem and amount of data necessary for

the implementation of real-life problems are usually large, a high capacity computing

facility will be used to test the different aspects of the algorithm(s). We will use a

high performance computing facility available at the computational science program

in the college of Natural Sciences.

3.4 Time Table

Phase I: (2011)

23

– Students will be recruited, the problem will be reformulated so that it can

suit to a team work at various level.

– Initiate a weekly research seminar

– Visit some selected public and/or business organizations and inspect how

they work

– Formulate mathematical models for the procedures they are working with

– Review literature relevant to the topics we will work on.

Phase II: (2012)

– Collect relevant data from selected public and business institutions (for

example, Ministry of Finance and Economic Development(MoFED), Min-

istry of Agriculture and Rural Development, Universities, Midroc Corpo-

ration, etc.) to validate the mathematical model developed.

– Develop algorithm(s) for bi-level programming problem with non-convex

lower level problem, with a single follower.

– Test the algorithm for a practical problem settings.

Phase III: (2013)

– Consolidate the data and information collected from the institutions.

– Formulate a workable mathematical model and propose algorithm that

yield a general purpose procedure which help decision makers in Ethiopia.

– Write the code and test the outcome of the algorithm with the aid of high

speed computers

– Finalize the study and write the final report.

3.5 Expected Outcome of the Research

∙ A well studied and organized procedure that may help the decision makers at

federal and regional offices in allocating resources will be proposed.

∙ A better solution method for multi-level problems with several decision makers

at each level will be proposed.

24

∙ An algorithm will be developed and is implemented on a computer so that

organizations (public as well as business) can use it efficiently.

∙ A scientific tool that enhance the capacity of consultancy services in the country

will be provided.

∙ Students will be trained at Masters and PhD level.

25

Component IV: “Mathematical

Epidemiology”

Semu MitikuDepartment of Mathematics, Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

E-Mail: [email protected] OR [email protected]

Keywords: dynamical systems, epidemiological models, infectious diseases

4.1 Introduction

Mathematical models in conjunction with computer simulations are proved to be use-

ful experimental tools for building and testing different theories, even when it seems

impossible to perform physical experimentations due to various reasons. It is now

customary to use mathematical models to analyse the spread and control of infec-

tious diseases. Especially since the foundation of the entire approach to epidemiology

based on compartmental models by Sir W. R. Ross, W. H. Hamer, A. G. McKendrick

and W. O. Kermack in the early 1900s the contribution of mathematical models to

health sciences start growing [11, 12].

In the last decade, epidemiological models have made important advances by recog-

nising the role of heterogeneous contact processes in the spread of infectious disease.

Nonetheless, existing models assume unchanged individual behaviour during out-

breaks of infectious diseases. In most of the epidemiological models describing the

dynamics of a disease, it is assumed that the average number of adequate contact

26

rates are either constant (in the case of the mass action incidence models) or formu-

lated as a function of the total population size only (in the case of standard incidence

models) [11]. Using these approaches many researchers produced several useful prop-

erties that describe the dynamics of the disease (at least for some relatively shorter

period of time since the spread of the disease) and its control mechanisms. Moreover,

various mathematical as well as epidemiological conclusions have been drawn using

this formulation. In practice, however, human population can learn from the risky

environment and change their own behaviour to protect themselves from contracting

the disease. This could change the dynamics of the disease if it persists for longer

time in the population. We believe that this situation is not adequately addressed in

almost all mathematical models studied for infectious diseases.

A mathematical model studied by Ying-Hen and Cooke [13] is used to conclude that

behaviour change and treatment of core groups can possibly eradicate the HIV/AIDS

pandemic. However, the specific effect of behaviour change and the self protective

measures were not explicitly analysed well in this paper. By taking the force of

infection as a variable, also dependent on behaviour change, Baryarama et. al. [14]

came into conclusion that a massive initial interventions are needed in order to contain

the epidemics. It is evident that behaviour modification is important in fighting

against any infectious disease, and therefore a large amount of investment is required

to help the population modify their behaviour. One can also argue that behaviour

adjustment can be observed as soon as the risk of contracting an infectious disease

gets high even though the actual investment in behaviour change campaign is not

that large.

As the morbidity and/or mortality caused by the disease increases, people get better

information about the diseases and start taking action to prevent themselves from

infection. As the awareness of the population about the disease increases they will

more probably take every available protective measure against the disease. This

will affect the contact rate per individual per unit of time in the dynamics, thereby

decreasing the incidence rate of the disease [15]. The intensity of the reaction of

the population depends on how much the disease is deadly or fatal. Especially the

reaction against the disease is stronger when the burden of the disease is very high.

That means, the prevalence of the disease may not pass certain threshold value, where

the amount varies from disease to disease depending on the condition how fatal it

could be.

27

Using utility analysis techniques Chen [16, 17] have shown that there is a unique en-

demic equilibrium prevalence exists when the basic reproductive number of a sexually

transmitted disease is strictly greater than unity. Moreover, Chen has also concluded

in [18] that the likelihood of eradicating an infectious disease through behavioural

changes depends critically on the amount of information individuals have access to.

This suggest that including self protective measures in the model will have a large

impact in the study of the dynamics of infectious diseases.

Thus, the aim of this project is to develop a theoretical framework that explains

how and why individuals change their behaviour in response to infectious disease,

thus providing information to improve the accuracy of models used to predict disease

spread.

A critical challenge for contemporary society is to understand and respond appropri-

ately to emergent infectious diseases such as H1N1 (swine flu) and SARS. To do this,

it is important to understand the processes that drive the spread and emergence of

infectious diseases. In the last decade, epidemiologists have made important advances

by recognising the role of heterogeneous contact processes in the spread of infectious

disease. Nonetheless, existing models assume unchanged individual behaviour dur-

ing outbreaks, although this assumption is known to be simplistic. Public health

and informal communication can impact on perceptions of disease threat, influenc-

ing individual behaviour and changing epidemiological dynamics. However, this full

spectrum has never been investigated comprehensively with the aim of informing pre-

dictive models. Missing from the synthesis is the contribution of social scientists with

capability in the investigation and modelling of individual and collective behaviour.

The team formed to work on this project has expertise in the areas of mathemat-

ical modeling and mathematical epidemiology, and an existing collaboration with

an international team developing epidemiological models of this type. The supervi-

sors will work with the student to develop a theoretical framework and collect data

via interviews and survey methods on social responses to infectious disease to verify

the theoretical results. Integration into existing models should lead to significant im-

provements in predictive accuracy and support simulations to explore targeted health

communication and make predictions relating to new diseases. This will allow health

bodies to design health communication and related interventions to maximise the

impact of limited resources and avert pandemic.

28

4.2 Objectives

∙ To develop a theoretical framework that explains how and why individuals

change their behaviour in response to infectious disease

∙ To analyze the impact of behaviour change in disease dynamics and give pre-

dictions for different types of disease models

∙ To train students in modeling and analyzing practical problems from public

health related issues and help them apply their knowledge in mathematics to

solve them.

∙ To train Masters level as well as Ph.D. level students specializing in Mathemat-

ical Biology.

4.3 Proposed Way of Conducting the Research

The first phase of the study is more of developing theoretical background and helping

students to review literature. In this regard, our plan is to recruit students both for

Masters level and Ph.D. level programs, establish research seminars serious dedicated

to topics in mathematical epidemiology to deepen their knowledge of the field and

help them understand the current scientific achievements in the field. We also need to

visit some public health centers and public health planning offices to understand the

way public health interventions are implemented. Collect necessary data, formulate

mathematical models for such procedures and test its efficiency is also a task of the

researchers in the meantime. This will require the follow up of the supervisory team

who will serve in consulting the students as well as directing the research.

Behaviour modification in human population is dependent on morbidity and mortal-

ity of a disease. Thus we modify the standard epidemiological models of infectious

diseases to incorporate a human learning effect to risky behaviours (or contacts to

infective agents). Using this model we investigate the behaviour of the dynamics of

an infectious disease in a given population, and the sensitivity of the dynamics to the

amount of effort to be made in the behaviour change campaigns to curb the infection.

Incorporating the informed group of population in the model may also help to clearly

see the contribution of education campaigns to protect people from infection.

29

4.4 Time Table

Phase I: (2011)

– Students will be recruited, the problem will be reformulated so that it can

suit to a team work at various level.

– Initiate a weekly research seminar and invite some researchers from abroad

to give short term courses.

– Visit some selected public health centers and public health planning offices

and inspect how they work

– Formulate mathematical models for the dynamics of various diseases using

behaviour modification as one possible parameter.

– Review literature relevant to the topics we will work on.

Phase II: (2012)

– Collect relevant data from the selected public health institutions to vali-

date the mathematical model developed.

– Analyse the mathematical model and estimate the parameters used in the

model.

– Interpret the results obtained in the language of epidemiology and recom-

mend best implementation strategy.

Phase III: (2013)

– Consolidate the data and information collected from the institutions for

various disease types.

– Formulate a workable mathematical model for various types of disease

models by incorporating a behaviour change cohort.

– Propose optimal strategy to implement public health education and phar-

maceutical interventions for various types of disease models.

– Finalize the study and write the final report.

30

4.5 Expected Outcome of the Research

∙ A well studied mathematical model for various types of disease models that will

take behaviour modification in to account will be proposed.

∙ Optimal strategy to implement public health education in conjunction with

pharmaceutical interventions (if any) in resource constrained communities will

be proposed for various types of diseases.

∙ Students will be trained at Masters and PhD level.

31

Component V: Algebra

Time-varying Linear Systems

Involving researchers

Berhanu Bekele and Tilahun Abebaw

Department of Mathematics, Addis Ababa University

P.O.Box 1176, Addis Ababa, Ethiopia

[email protected], [email protected]

Keywords: Algebraic analysis, ordinary differential equations with rational/polynomial

coefficients, behavioral approach, module theory, skew fields.

5.1 Introduction

The analysis and synthesis of linear time-invariant systems have, to a great extent,

dominated the efforts of control theorists and control engineers. As a result a large

body of literature on these subjects presently exists compared to linear time-varying

systems. The reason for this domination is that it is an important case with a rich

structure. The theory can be called well developed in most respects by now, especially

in the case of ordinary differential equations.

A theory of linear time-varying systems is still needed because time-varying systems

appear in many fields, for example, in the control of modern aircrafts and space

crafts where increased accelerations and velocities induce parameter variations. The

flight of a rocket where large amounts of fuel are burned very rapidly is a time-

varying problem. In electronics, parametric amplifiers, and microphone transmitters

containing a variable resistance are also time-varying systems. They can also be used

32

for approximation of time invariant non-linear systems. Recently, there have been

several advances, from different angles, to tackle the time-varying counterpart [20],

[23], [25], [26], [27], [31], [32], [33], [34].

The seminar paper (Oberst, 1990) established a link between the algebraic analysis

approach and the behavioral approach to systems and control theory, introduced by

Willems in the 1980s(Willems, 1991). More precisely, Oberst introduced a categor-

ical duality between the solution spaces of linear partial differential equations with

constant coefficients and certain polynomial modules associated with them.

The property of some signal spaces (e.g. the space of smooth functions or the space

of distributions) when considered as a module over the ring of differential operators,

namely, the injective cogenerator property plays an important role in the study of

time-varying systems. This property makes it possible to translate any statements

on the solution spaces that can be expressed in terms of images and kernels, to an

equivalent statement on the modules. Thus analytic properties can be identified

with algebraic properties, and conversely, the results of manipulating the modules

using (computer) algebra can be re-translated and interpreted using the language of

systems theory. This duality is widely used in behavioral systems and control theory.

The generalizations of the theories that are developed for time invariant systems to

varying coefficients are not at all straightforward. This is not so much due to the

technicality that the rings of differential operators to be considered lose their Com-

mutativity when passing from the constant to variable coefficient case. The structural

properties of linear time-varying systems with rationally-varying coefficients, both in

the continuous- and discrete-time case, are studied in [21]. Systems of linear differen-

tial and difference equations are studied using algebraic tools such as module theory

and homological methods. The realization theory of continuous-time time-varying

linear systems and matrix fraction description of a transfer matrix defined over a

skew field are also studied. For continuous-time time-varying systems, Kamen [28]

has obtained several results for systems of the form , in which and are defined over

skew noetherian rings and is monic. Using this result a transfer matrix for rationally-

varying systems that are given in state space representation is obtained in [21]. The

realization problem is also addressed. Minimality of a realization is related with

the concepts of controllability and observability, as in the time invariant case. The

notions of coprimeness, irreducibility of matrix fraction descriptions and row proper-

ness are also discussed. As a result, it is showed also the existence of row proper

33

representations, and input-output representations with proper transfer matrix. In

application, however, there are systems with polynomial coefficients. Studying such

systems will address the controllability and observability problems of systems with

polynomial coefficients. Moreover, the state space realizations, which are well known

in the time invariant case, will be studied in polynomial-varying settings.

The smith and McMillan forms of a matrix play an important role in realization

theory. An algorithm for finding smith and McMillan forms of a matrix are developed

for the time invariant case. The generalization of the McMillan form to varying

coefficients is not yet done. In this study we consider the generalization of McMillan

form to the time-varying cases.

5.2 Objectives

∙ To develop an algorithm for finding McMillan form to time-varying cases.

∙ To study systems with polynomial coefficients, that is, to generalize results from

the time invariant case to systems with polynomial coefficients.

∙ To implement the algorithm in control toolbox.

∙ To train Masters and PhD level students specializing in algebraic system theory.

5.3 Proposed Way of Conducting the Research

The first part of the study will be devoted to the study of systems of linear differen-

tial and difference equations with Polynomial coefficients using algebraic tools such

as module theory and homological methods. To have a clear understanding of the

problem, the students will first review literatures that are related with time invariant

systems and time-varying systems with rational coefficients. Then, generalize those

results to our case by making the necessary assumptions and modifications.

34

The second part is concerned with the development of an algorithm or procedure to

find the McMillan form of a matrix with time-varying entries, particularly, for the

case of rational and polynomial entries.

The final phase of the study will be the implementation of the algorithms that are

developed and testing them with particular problems.

5.4 Time Table

Phase I: (2011)

∙ Enrollment in the masters and PhD programs in Algebra of two junior staff

members of the department.

∙ Problem formulation for the PhD program.

∙ Literature review.

Phase II: (2012)

∙ Work on the problems to get generalizations of results form the time invariant

case to the time-varying case.

∙ Develop algorithm for the computation of the McMillan form and test it.

∙ Present their findings in a seminar form in a weekly base.

35

Phase III: (2013)

∙ Continue the research to obtain generalizations of more results form the time

invariant case.

∙ Prepare publication of the results in the form of Journal papers.

∙ Implement the algorithms in control toolbox.

∙ Finalize their work and write the final report.

5.5 Expected Outcome of the Research

∙ An algorithm to compute the McMillan form of a matrix will be developed and

is implemented in control toolbox.

∙ Duality between time-varying systems (particularly with polynomial coefficients)

on the one hand and modules over the ring of differential operators on the other

will be established and used to check controllability and observability of the sys-

tem.

∙ Students will be trained at masters and PhD level.

36

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